26.5.15 I have just written an obituary of John Nash and his wife Alicia for the U.K. Guardian: http://www.theguardian.com/science/2015/may/25/john-nash

It describes Nash’s contributions, not only to game theory, but also to mathematics.

Here is the original draft:

John Nash, who has died in a taxi accident at the age of 86, was a mathematicians’ mathematician. Subject of the film *A Beautiful Mind*, his name will forever be associated with the *Nash equilibrium* – a position in a situation of competition or conflict where both sides have selected a strategy, but where neither side can then independently change their strategy without ending up in a less desirable position. Such positions are very common in our everyday social interactions, not to mention the interactions of business people, politicians, and nations. Nash earned his early reputation, and his 1994 Nobel Prize in Economics, by proving mathematically that there is at least one Nash equilibrium lying in wait to trap us in *every *situation of competition or conflict where the parties are unwilling or unable to communicate.

Nash had arrived at Princeton University in 1948 to study for a post-graduate degree in mathematics, bearing a laconic one-line recommendation from his previous professor Richard Duffin “This man is a genius.” He proved his genius within two years by publishing what is surely the shortest paper ever to win its author a Nobel Prize. Called “Equilibrium Points in N-person Games,” it was less than a page long, and contained just 317 words. It was a major contribution to the burgeoning field of game theory, whose foundations had been laid just a few years earlier by mathematician John von Neumann and economist Oskar Morgenstern at the Princeton Institute for Advanced Study in their seminal work “Theory of Games and Economic Behaviour”.

Game theory provides a mathematical analysis of the strategies that we use every day in our interactions with other people. With Nash’s insight, it provides a focus for understanding the roots of many problems in conflict and the failure of cooperation that we face today.

Nash himself, however, did not regard his contribution particularly highly. He was more concerned with tackling basic problems across the whole range of mathematics. Even as an undergraduate he had produced an independent proof of Brouwer’s fixed point theorem – the theorem that tells us that, no matter how much we stir a cup of coffee, there will always be one small bit that is just where it was before we started stirring. In 1954 and 1956 he produced the two remarkable Nash embedding theorems, which are difficult to explain to non-mathematicians (they prove that “every Riemannian manifold can be isometrically embedded into some Euclidean space”), but which provided the basis for much subsequent mathematics. One (very simplified) way to illustrate their main point is that, no matter how you bend a piece of paper with lines drawn on it, the lines will always have the same length. It sounds intuitively obvious, but the proof was difficult, and the theorem has led to many counter-intuitive conclusions.

Nash’s mental illness that was the main subject of the film began to make its appearance in 1959. He could be quite funny about it in later years, and said to one interviewer “Mathematicians are comparatively sane as a group. It is the people who study logic that are not so sane.” But the reality was not so funny. His diagnosed paranoid schizophrenia led to the break-up of his marriage (although he re-married his wife Alicia in 2001), the loss of his job, and nearly the loss of his Nobel Prize when members of the selection committee were worried that he might never be able to pursue serious research again.

But carry on he did. In between bouts of hospitalization that have been well documented, and gradual recovery after many severe episodes in the 1970s, 1980s, and early 1990s, Nash continued to make contributions to many area of mathematics, especially in the solution of partial differential equations, which describe how several factors affect each other when all are changing simultaneously. Such equations are basic tools for engineers and for people developing computer models of our economies and our societies, but they are notoriously difficult to solve. For his contributions Nash was awarded the Abel Prize by the Norwegian Academy of Arts and Letters on May 19th. He shared the prize with Louis Nirenberg, and talked of his plans for work in the fields of cosmology and general relativity (when he first arrived at Princeton in 1948, he had reputedly demanded an interview with Einstein so that he could explain where the general theory of relativity was wrong). It was after returning to the U.S. with Nirenberg, who farewelled them at Newark airport, that he and his wife Alicia took their fatal taxi ride.

John Forbes Nash Jr. was born in Bluefield, West Virginia on June 13^{th}, 1928. His father John Forbes Nash was an electrical engineer, and his mother Margaret Virginia (née Martin) had been a schoolteacher. After taking advanced mathematics courses at a local community college during his final year of high school, he attended the Carnegie Institute of Technology (now Carnegie Mellon University), initially majoring in chemical engineering. He switched to chemistry, and then to mathematics, graduating in 1948 with B.S. and M.S. degrees in mathematics, after which he accepted the John S. Kennedy fellowship to study for a Ph.D. at Princeton University.

On completion of his thesis (just 27 pages long!), he was hired by M.IT., initially as an instructor in the mathematics faculty, and then in 1958 as a tenured professor. Early in this time he had a relationship with Eleanor Stier, and in 1953 they had a child, John David Stier. In 1957 he married Alicia Lopez-Harrison de Lardé. He was also a consultant for the RAND corporation from 1950 to 1954, but lost his position (even though all charges were dropped) after he had been arrested for indecent exposure in a police entrapment operation in Santa Monica, California.

In 1959 the symptoms of the paranoid schizophrenia that was to plague much of the rest of his life made their appearance. He resigned his position at M.I.T. in the Spring of 1959, and Alicia had him admitted to McLean Hospital for treatment. Their son John Charles Martin Nash, who also became a mathematician, was born on May 20^{th} of that year.

The stress of Nash’s illness took its toll, and he and his wife were divorced in 1963. After various periods of hospitalization he was finally discharged in 1970, but continued to struggle with paranoid delusions over the next two decades. In an interview in 2004, he said “I was a long way into mental illness before I heard any voices. Ultimately I realised I am generating these voices in my own mind: this is dreaming, this is not communication. This is coming from an internal source, not from the cosmos. And simply to understand that is to escape form the thing in principle. After understanding that, the voices died out.”

John Nash was awarded the John von Neumann Theory Prize in 1978, the Nobel Memorial Prize in Economic Sciences in 1994, the Leroy P. Steele Prize in 1999, and the Abel Prize in 2015. In 2001 he remarried Alicia, who died in the same car crash on May 23^{rd}, 2015. They are survived by two sons: John David Stier and John Charles Martin Nash.

IMAGE: Princeton University archives